Talk:Pseudo-range multilateration

Multilateration vs Trilateration
National labs for metrology in Germany (PTB), UK (NPL), US (NIST) and Japan (AIST) refer to multilateration as determining target coordinates by measuring the distance to the target from multiple locations (by whatever curvilinear line-of-measurement method that works within the geometry of the problem). While multilateration may have originated regarding one measurement technique (TDOA) and may still be considered to hold to that narrow definition within specialized disciplines, it is now being used in the broader sense above worldwide. The common use of the term multilateration for interferometers and other distance measurement devices should be covered in the article. What I think makes most sense is that trilateration refers to 3 stations and multilateration refers to an ambiguous number (3 or more), so I think that is where the terms have headed towards.

Here are just a few of many articles from work by the national labs and others using the term multilateration for laser interferometry, which is distance measurement, and NOT a TDOA method.

http://secure.theengineer.co.uk/Articles/305424/Laser+TRACER+calibration.htm

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4K-4F490P6-1&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=d97133bda6f334a4296adcde6d667509

http://www.slac.stanford.edu/econf/C04100411/papers/054.PDF http://www.ptb.de/de/org/5/nachrichten5/archiv/2006/wirtschaft/abteilung5_18.htm http://www-pnp.physics.ox.ac.uk/~licas/page_talks/IWAA2008/IWAA2008_John_Dale.pdf http://www.ptb.de/en/publikationen/jahresberichte/jb2000/jb5e.pdf —Preceding unsigned comment added by 70.102.119.35 (talk) 22:42, 12 February 2009 (UTC)


 * I agree. The article about TDOA / Multilateration, need to incorporate the wider use of the term. I also think it should be noted that Trilateration is basically a special case of Multilateration. I hope someone with enough knowledge, time and wiki-experience could rewrite this article. Haakoo (talk) 11:10, 29 July 2009 (UTC)

Multilateration VS hyperbolic positioning
Multilateration uses spheres, as trilateration uses circles. DTOA uses hyperboloids and is hyperbolic positioning and is an completely different thing, not the same as multilateration. In http://www.aviationtoday.com/av/issue/cover/9891.html is described how multilaterition uses spheres and NOT hyperboloids. For the multilateration calculations it is not important how the size of the spheres is acquired. —Preceding unsigned comment added by 92.66.245.162 (talk) 16:22, 25 June 2008 (UTC)
 * This comment is somewhat anachronistic, the usage of the terms had widenen in the last decade. The restricting of multilateration to spheres is wrong. Multilateration applies, wherever the number of measurements is over determined compared to the minimum requirements. Hence multilateration is also state of the art for planar or cubic problems, not just for spheres. Using DTOA from the viewpoint of measurement applies as well to phase measurements. There is no exclusive restriction of the term to hyperbolic locating. Finally, the common term 'positioning' is misleading, as no position change is effected, all topics discussed touch 'locating' which is determining a location in a known system of coordinates or under equal entities which no fixed systemof coordinates.Wireless friend (talk) 07:04, 12 November 2008 (UTC)
 * the article currently seems to put too much emmphasis on the hyperbolic thing and doesn't make any mentions of spheres or circles --TiagoTiago (talk) 07:01, 30 May 2009 (UTC)

Math errors
I think that the math in the basic formula is a bit confused. (x, y, z) is given as the unknown location of the emitter, but then also as the location of the central transmitter. Different variable names should be used.

While I mention this, I would also suggest that the mathematical discussion be fleshed out a bit more by listing what should be minimized in the optimization.

Mark (talk) 19:46, 29 April 2008 (UTC)

Decca
I am no expert in this field. But I would like clarify a small matter.

My understanding of the situation (and its 20 years since I did my exams) is that the LORAN hyperbolae were based on time differences, but that Decca and Omega were both based on phase differences and NOT time differences.

If this is the case then one of the following actions should be taken:

1. The reference to multilateration should be removed from the decca article; or 2. The definition on the multilateration page should be changed to include phase or time difference


 * Again on anachronisms: The sense of the words is correct, phase is an entity of time, as is frequency. Both terms apply as well to continuous wave transmissions as to pulse coded transmissions. The resolution of frequency and pahse delivers enhanced resolution, but not any different topic.Wireless friend (talk) 07:08, 12 November 2008 (UTC)

Response
I have modified the article to include mention of the phase approach, and also modified the DECCA article to say "approach similar to multilateration". In fact, as the DECCA transmissions are continous wave, I think it is correct to describe the approach as multilateration. Phase-difference and time-difference are essentially the same thing with a narrow band source. Paul 06:42, 26 December 2005 (UTC)


 * This does not make sense to me. Either Multilateration includes phase difference, in which case the description "approach similar to multilateration" is incorrect for Decca, or it excludes phase difference, in which case Decca is not multilateration. The NPL describes it as "Multi-lateration is a way of measuring the position of a target, or targets, relative to several fixed measuring stations" at http://www.npl.co.uk/length/dmet/science/multilateration.html. This appears to include phase difference. --SC 22:17, 29 December 2005 (UTC)


 * I think that the confusion originates from the fact that this entry describes Multilateration as "the process of locating an object by accurately computing the time difference of arrival of signals from three or more locations to that point". Having read around a bit more it appears to me that Multilateration measurements do not have to be by TDOA (although they presumably are usually). This article also duplicates the principles already covered in the trilateration article. I propose the following actions:
 * Rewrite multilateration removing dependence on TDOAs, whilst recognising that TDOAs are often used.
 * Remove duplication between multilateration and trilateration. (The Trilateration article is probably a better place to describe the principles, since it is an easier concept to grasp.)
 * Reduce emphasis on Decca in both articles, since it is obsolete and (I think) that GPS is hyperbolic also, so that would be a better reference
 * Add description of what multilateration adds to trilateration. This is unclear to me.

--SC 08:19, 30 December 2005 (UTC)


 * The problem with using GPS as an example is that it is very difficult to plot GPS hyperbolae on a chart or map. For a start they are 3-dimensional. They are time-difference based and the stations are in orbit above us.


 * The nice thing about Decca and/or Loran is that the hyperbolae can be (and often are) overlayed on a navigational chart and the readings plotted as lines of position to give a position fix. It was certainly seeing overlayed hyperbolae printed on charts that made it all "click" for me all those years ago. I seem to remember our Electronic Nav lecturer using a North Sea chart with the lattices from 3 different Decca chains printed thereon and the positions of at least 2 master stations and numberous slave stations highlighted along with the baselines. It just all fell into place.


 * Now trying to do something similar with GPS LOP's, that would be fun.Frelke 09:44, 30 December 2005 (UTC)


 * Again I am not claiming expert status here, but isn't multilateration just a generic form of trilateration, i.e tri = 3 and multi = >1. I am guessing now, but I think we are headed for merge here.

Frelke 09:49, 30 December 2005 (UTC)

Clarification
I'm struggling to understand the confusions here. Multilateration is the determination of location using multiple receivers. Trilateration is with exactly three. Both use TDOA to determine the intersection of 2 (with trilateration) or N-1 (with multilateration using N receivers) hyperboloids. Hence the term hyperbolic positioning.

TDOA is usually measured by measuring time of arrival directly, but equivalently can be determined by measuring phase difference - but only if the signal is narrowband.

If anything needs to be merged, then trilateration should be merged into this article, as trilateration is just a special case of multilateration.

The article already says all of this. Paul 21:37, 18 January 2006 (UTC)

It's mixed up
Quote «Multilateration, also known as hyperbolic positioning...»

1. Multilateration (including trilateration) is based on estimation of the time of arrival (TOA). 2. Hyperbolic positioning is based on estimating the time difference of arrial (TDOA). 3. Doppler positioning is based on estimating the dopler shift of the satellite signal.

These are three (3) main (and different) types of radionavigation. See, for example, book Global Positioning System by Pratap Misra and Per Enge (page 12, chapter 1.2 "Methods of Radionavigation").

Kender 05:37, 18 January 2006 (UTC) Stanford, CA


 * Now I'm totally confused. So what is the difference between types 1 and 2 above ?
 * Is there a difference between is based on estimation of the time of arrival and is based on estimating the time difference of arrial. I think we have already agreed here that Hyperbolic positioning can be based on more than just time difference. It can be based on phase difference (as in the case of Decca and Omega). So can we clarify what Misra and Enge mean (presumably these definitions are theirs).


 * My biggest problem with all this is that I still believe that whatever multilateration is, the only difference between it and trilateration is the number of position lines used, tri having 3 p/l's and multi having an unspecified number.


 * Frelke 07:35, 18 January 2006 (UTC)


 * Simple rule of thumb: "Lateration" is a type of positioning system that uses distances from reference points to calculate the location of a signal source based on the properties of the hyperbola. "Angulation" is a type of positioning system that uses angles from reference points to calculate the location of a signal source based on the properties of the triangle. "Trilateration" and "Triangulation" are special cases of the "lateration" and "angulation" methods, that refer specifically to using three known quantities to determine two unknown quantities using one of those two methods of calculation. Multilateration, then, refers to the general case of using N+1 known quantities to generate N unknown quantities using distance measurements.


 * 121.75.101.150 (talk) 00:38, 7 July 2012 (UTC)

It's all based on the difference between TDOA and TOA.

Since historically TDOA was used for navigation before TOA, let's start with TDOA. Consider two transmitters spaced far apart and one receiver (or user). Each of the transmitters sends a pulse at the same time – they are synchronized. A user first receives a pulse from transmitter 1 then from transmitter 2. The delay between the pulsed is TDOA. TDOA=TOA1-TOA2 (Because of the clock bias, user doesn’t even know the TOAs.)  In 2D TDOA from one pair of transmitters puts a user on the hyperbola; hence TDOA systems are also known as hyperbolic systems. To estimate the position the user needs at least two pairs of transmitters (two TDOAs). Each of the pairs will produce a hyperbola and the user position is at the intersection of these hyperbolas.

Next, consider one transmitter and one receiver. One pulse arrives to the receiver. There is no TDOA, because you need two pulses to produce the difference. However, if both receiver and transmitter are somehow synchronized to common time, TOA can be estimated. In 2D TOA from one receiver puts a user on a circle. To estimate the position the user needs at least two transmitters (two TOAs).

Quote «The problem with using GPS as an example is that it is very difficult to plot GPS hyperbolae on a chart or map.»

GPS doesn't produce hyperbolas, because it's not a TDOA system. It's a TOA systen, and the LOP in 3D is a sphere. Kender 08:17, 18 January 2006 (UTC) Stanford, CA

the above is mixed up!
The statement "multilateration is based on estimation of the time of arrival (TOA)" above is incorrect. Multilateration uses the time-difference of arrival of a pulse between two sites (see, for instance, or ). Absolute time of arrival is not required, and not even measured in systems such as VERA. The explanation above of two transmitters and one receiver is correct - but just the reciprocal case of what I just described. Both are TDOA. So, in terms of the list above, both 1) and 2) use TDOA, and both can be called multilateration or hyperbolic positioning. Incidentally, in my professional life I work on this technology, and the term multilateration is commonly used in the way described in this article.

Paul 06:16, 19 January 2006 (UTC)

Merge
I am proposing to merge this page with trilateration, this being the more general case. I think that the other article is actually the better article and so would intend to keep the vast majority of it.

See discussion page for vote.

Frelke 07:42, 19 January 2006 (UTC)


 * I would agree - but see my more detailed comments on discussion page. Care is required. Paul 11:32, 19 January 2006 (UTC)


 * Frelke, do you think that trilateration is a particular case of multilateration? Kender 04:15, 23 January 2006 (UTC) Stanford, CA


 * I think that it is, but I am not an expert. I just studied hyperbolic navigation many years ago and my brain still works. Frelke 11:18, 23 January 2006 (UTC)

Loose ends?
The article is much improved after the recent activity. I presume it is correct now ;-) If so, there appear to be a few loose ends to tidy up as the Multilateration article says that Decca used Multilateration but the Decca article says that is used "an approach similar to multilateration". Does this mean now that the Decca article (and others) needs updating? --SC 22:43, 27 January 2006 (UTC)


 * Again, I am not the expert, rather the person who stirs the bucket to get a reaction. I remember that comment going into the Decca article and thinking myself at the time that it sounded 'uncomfortable'.


 * I suggest that it be changed to "...a multilateration-based approach..." unless anyone has a better alternative. Frelke 23:25, 27 January 2006 (UTC)


 * I just updated the DECCA article seeing as it was my unfortunate wording in the first place. I simply changed it to "also known as multilateration". --Paul 12:29, 30 January 2006 (UTC)

GPS
GPS works by receiving several known signals in 1 location, not by sending one signal, and receiving in several locations. So I'd say that GPS is not a valid example (but location determining in cellphone networks can be). Am I right?

NavigationGuy (talk) 16:56, 31 December 2018 (UTC) No, you're wrong. A-n-y Radio NAVIGATION system receives multiple signals at one location (often, a vehicle). For a SURVEILLANCE system, it's the opposite. The vehicle transmits and multiple locations receive the same signal.

One can 'do' navigation using multiple true ranges (e.g., aircraft DME/DME) or multiple pseudo ranges (ranges with a common offset, as the time of transmission is not known to the vehicle). Examples are GPS, LORAN-C, Omega or DECCA. Similarly, one can 'do' surveillance using multiple true ranges (FAA ERAM sort of does this) or multiple pseudo ranges (e.g., WAM, ASDE-X).

NavigationGuy (talk) 14:42, 16 December 2018 (UTC)


 * I would rather say that where you do the measurement is of no important. The basic principle is the same. You measure the difference in distances between several references and an unknown position. If the references or the "locator" sends the signals is irrelevant. (Håkon K. Olafsen) 193.157.188.206 14:31, 25 March 2007 (UTC)

WHAT is measured matters. In virtually all MLAT systems, TOAs are actually measured (TDOAs are then calculated). In error analysis, TOAs are usually assigned statistics -- not TDOAs.
 * When the stations are fixed to the earth, one can subtract n+1 TOAs to get n DTOAs and solve for the user's position in n dimensions.
 * When the stations are satellites, one processes the n+1 TOAs simultaneously to get the n DTOAs and the 'user clock offset' (assuming navigation).

NavigationGuy (talk) 16:15, 20 December 2018 (UTC)


 * At the conceptual level, GPS works like LORAN-C, Omega or DECCA. With each system, the navigation receiver detects and process signals from multiple synchronized transmitter stations to obtain the differences in ranges to the transmitters.  Those range differences (for N stations there are N-1 independent differences) are used to derive the navigators position (e.g., latitude, longitude and altitude).  GPS is a three-dimensional system, so signals from at least four stations (satellites) must be received.  LORAN-C, Omega or DECCA are/were all two-dimensional systems, so signals from at least three stations are needed.  (Early LORAN-C receivers and DECCA receivers could only process the minimum three signals.)  Passing from the conceptual realm to practical implementation involves selecting the signal frequency and modulation technique.  GPS and LORAN-C basically measure time-difference, while Omega and DECCA measured phase difference.  But in either case, the time or phase difference is equivalent to a range difference.  — Preceding unsigned comment added by NavigationGuy (talk • contribs) 12:53, 28 October 2014 (UTC)

Three receivers - one additional TDOA?
I think the Principle section should be reformulated in this way:
 * For two receivers we get one hyperboloid (either one of the pair, depending on the sign of TDOA). Adding THIRD receiver brings in TWO additional hyperboloid, because for each pair of receivers there is one (assuming the three receivers are not on one line). So two of these hyperboloids give us a curve while the third one should pin-point the location precisely. Am I right?
 * MiShogun 08:59, 31 July 2007 (UTC)
 * For two receivers you get one hyperboloid. Because you have one TDOA. Adding a THIRD receiver gives you three TDOA, but they are dependend on each other. TDOA3 = TDOA1 + TDOA2. Because they are dependend you still have actually only two nonrelated parameters. To be able to describe any point in space you need at least 3 nonrelated parameters. The TDOA3 does not give any extra information. The first two paraboloids intersect in a closed curve the third intersects the same curve as wel. You need a Fourth receiver to be able to pin-point. And then there are still 2 solutions, two pin points.
 * Or for GPS you need the reception of 4 satellites to do 3D pinpointing.
 * Crazy Software Productions 17:32, 5 September 2007 (UTC)
 * You're almost right. With 4 reference points and known difference in distance between the references and the unknown position, it is possible to find one unique solution in space (3D). You can also see Trilateration for this.
 * With 4 reference points (4 receive times, so 3 independend differences), there are two solutions (x1,y1,z1,t1) and (x2,y2,z2,t2). The second point is not stable, moving fast and the solution can have a 'large' time offset from the actual time. So although there are two solutions, it's easy to bin one of the solutions. But somewhere in space and somewhere in time there is a point with exactly the same time differences as the 'actual' point. The algoritm show at trilateration starts from knowing distances and at the start of the calculation they are not available..Crazy Software Productions 14:38, 8 September 2007 (UTC)
 * It's obviously something I'm not getting here. Where does t1 and t2 come from? With four reference points, there are one unique solution to a localisation problem in 3D (as long as the references are not in a line, in the same spot and a few other special cases).
 * With four reference points there are two solutions. With almost all configurations there are two solutions. One of the solution is not stable, far removed from earth and with a time-error which can be huge, but there is a second solutions for the four reference points. If you are working with the model with the spheres, you normally alter the size of the spheres with the same amount until they intersect at one point. But if you keep changing the size of the spheres (often to very large) they will intersect in another point, often far removed from the first point and with spheres of a larger size. (The size of the spheres represent time timing (error)). —Preceding unsigned comment added by Crazy Software Productions (talk • contribs) 13:44, 18 September 2007 (UTC)
 * Multilateration as I've understood and used it, does not necessary involve TDOA measurements. It's the mathematical calculation of the localisation when the difference in distance (time transfers to distance) from the unknown position to the references are known. I think this article needs to be completely rewritten, and multilateration treated as a mathematical concept, and be distinguished from TDOA measurements.
 * GPS with a quartz clock is based on TDOA measurements, just because no absolute time is available. I try to avoid the terms Trilateration and Multilateration because they are not completely clair. TOA and TDOA are better defined. I would think that trilateration (as described in a lot of places) is TOA put at some places this is contested. TDOA gives hyperbolic shapes (althought this does not have to present itself in the calculation) and that this is often refered to as Multilateration.
 * My point is that trilateration and multilateration is two techniques used to solve a localisation problem when you got a set of data. The data used in multilateration is basically the difference in distance to certain reference points. You do not necessary find this difference with TDOA. If you measure the distances to the references, multilateration can ease the calculations and help improve the accuracy by suppressing errors in the measurements.
 * The time / distance measurements does not even have to be synchronous, as long as the unknown position is not moving, or moving very slowly.
 * If the time /distance measurements are not done very close together in time, then the accuracy of the clock becomes very important. With a very good clock, you could do position calculation with only one satellite (in four different places), you only need four measurements, but a extremely accurate clock is needed. (The drift in a very good quartz clock is 10 meters each second. The drift in a 'normal' quartz clock is 500 meters each second. So if the measurements are done within a second, with a normal clock, the position calculation is not very accurate anymore).Crazy Software Productions 14:38, 8 September 2007 (UTC)
 * This is one of the problems by having a to close relationship with TDOA and multilateration. You can used the same equations as multilateration, except the 1/c part if you know the distances. TDOA needs multilateration, multilateration does not need TDOA.
 * Haakoo 05:23, 17 September 2007 (UTC)
 * Haakoo 06:10, 7 September 2007 (UTC)
 * Thanks for clearing this up. Took me a long time to get what you are saying. As I get it now for multilateration there could be other methods than time differencing. (At the moment I can think of none). Part of the confusion I think comes from that so many people try to explain tri and multi as confined to certain numbers. I think that the tri is from triangle, and that three points form a triangle so for triangulation two known points can define the third point, equivalent for trilateration again based on triangles. And not that three known points should be given. Multilateration for me at least suggests a number (maybe more than tri?), but could also mean multi triangles, or multiple sources for one parameter (difference of two signals/distances ?). So it took me quite some time to get you were talking that TDOA also defines how we get the info and multilateration just does not specify how we get the info. I don't know any application of multilateration which is not based on TDOA, so I thought of them as equivalent (or almost the same). (Sorry My mistake.).
 * Do you have an example of multilateration which is not based on TDOA?
 * Crazy Software Productions 18:17, 18 September 2007 (UTC)
 * The name is confusing, and I had a good time trying to track down different localisation algorithms when I was working on my master thesis. I'm not sure why they are called multilateration and trilateration, might be because the equations used in multilateration are easily used in an over determined system. If you take a look at my master thesis I (we) consider WSNs where the nodes can measure the distance between themselves, and then creates a grid of locations from this information. The technique used to estimate the distance (range) between nodes is found in my friends master thesis. It is not a working system, but you get the idea.
 * My master thesis "Wireless Sensor Network Localisation Strategies" is available from: http://wo.uio.no/as/WebObjects/theses.woa/wa/these?WORKID=60422 Chapter 4.7.3-4 show the mathematics for "multilateration" and an over determined case. And chapter 3.4 discuss data acquisition.
 * Nikolaj's master thesis "High Precision Ranging in Wireless Sensor Networks" is available from: http://wo.uio.no/as/WebObjects/theses.woa/wa/these?WORKID=58956 Chapter 3 is probably most interesting.
 * I guess my biggest problem with the multilateration article is that Hyperbolic positioning and TDOA points diretcly to multilateration, and if this is to continue, multilateration needs to be rewritten. Or we can have the mathematical calculations and explenations in Hyperbolic positioning and let multilateration be (HP+)TDOA. I'm not sure what's the right thing to do.
 * Haakoo 07:01, 22 September 2007 (UTC)

Major clean up
This article with it's redirects really needs a major clean up. The Multilateration article should either be similar to the Trilateration article, without the strong connection to Time difference of arrival / TDOA or Hyperbolic positioning should not redirect here. I do not have any good references for the present norm in literature. My opinion is that the Trilateration article is better than this one, causing less confusion. It basically ends up with a discussion of the meaning of "Multilateration", and how it should be used.

If multilateration is considered to be the process of doing a TDOA and the estimate the position, Hyperbolic positioning should not be redirected here. If multilateration on the other hand is the process of estimating a position based on a given set of data (difference in distance to reference points of known position) then TDOA should not redirect here and the article rewritten.

TODA deserves it's own article just as Time of arrival, and this article should be rewritten without the strong connection to TDOA. As I've said earlier, multilateration does not need TDOA data to estimate a position.

Haakoo 02:56, 26 September 2007 (UTC)

Terrestrial radionavigation
The article describes 3D-space case, where distances are straight lines, but no words are said about spherical case of terrestrial radionavigation (e.g. Loran-C), where distances are calculated using haversine formula. Unomano 07:06, 11 October 2007 (UTC)

That external link

 * "Multilateration is a proven technology that has been in use for many decades" ... "Using advanced computer processing techniques" ... "Multilateration requires no additional avionics equipment"

That's not a maths page, that's a sales page. Hardly relevant in an encyclopedia article describing the technology? Ojw (talk) 14:34, 15 October 2009 (UTC)

More Math
I have expanded the 3-D solution section and added sections defining the geometry and some stuff on TDOA measurement. In general - lots of additional math that should lead to a solution. The "Intro" and "Principal" section were not changed. It will take some thinking and homework to sort-out the swirling questions of the various relations (or lack of relations) between TOA, TDOA, trilateration, the assorted meanings applied to the word multilateration and the example systems mentioned in the article. —Preceding unsigned comment added by TinyPebble (talk • contribs) 01:30, 20 March 2010 (UTC)

Typo?
I think there's a typo in the article. Under "3-D Solution" in the 4th paragraph: "Improving accuracy with a large number of receivers can be a problem for devices with small embedded processors because of the time required to solve several simulatious..." What does the author mean by "simulatious"? Did he mean "simultaneous?" Krenzo (talk) 00:07, 19 December 2010 (UTC)

Reference for 3-D Solution section
None of the other articles that I've found seemed to include the trick for removing the 2 R0 term. Is this somewhere in the Bucher/Misra paper that I just missing, was it your own work, or some other source? It's a very nice trick and I want to ensure that I'm referencing it correctly in the biblography for my thesis. — Preceding unsigned comment added by 205.167.170.20 (talk) 19:45, 22 June 2011 (UTC)

Sorry for the 6 month delay in responding. The math trick is original and it is appropriate to reference the Wikipedia article. It is nice to see open license info in action. I am sure you have seen that many people get ruffled feathers with references to a URL page in Wikipedia because anyone can change it. I would suggest including the date with the URL reference. That will allow people to see the referenced math and text before any future changes by looking at the history page. TinyPebble (talk) 06:25, 7 December 2011 (UTC)

Inaccurate Cross-Correlation Plots
Please examine figures 3a, 3b, and 3c. The two signals in each figure appear to be separated by about 2 time units, but the cross-correlation plot indicates a separation of 5 time units. I feel the peaks on the cross-correlation plots should be closer to 2. Please let me know if I am wrong on this issue.146.165.84.65 (talk) 18:12, 25 July 2011 (UTC)
 * I Agree! I also came here to point that out. Things like this bother me. Unfortunately I currently don't have software to create such a plot. 84.137.186.78 (talk) 21:20, 20 September 2012 (UTC)
 * Yep. I came here to report exactly the same error. --ps — Preceding unsigned comment added by 99.109.150.114 (talk) 05:11, 23 July 2013 (UTC)

All of you are correct. The time shift between P0 and P1 needs to be 5 time units to get a cross correlation peak at 5. Sorry it took me so long to fix the graphs. TinyPebble (talk) 04:04, 13 June 2015 (UTC)

Inconsistencies in the number of linear equations
The article states: "Use equation 7 to generate the four constants Am,Bm,Cm,Dm from measured distances and time for each receiver 2 ≤ m ≤ N. This will be a set of N homogeneous linear equations."

That would actually leave us with N-1 equations. For example, say you have 4 receivers (P0, P1, P2, P3), we would have N=3. If we generate an equation for each receiver 2 ≤ m ≤ N, this would leave us with 2 equations (m=2, m=3). Unless there is another equation that I missed. — Preceding unsigned comment added by 192.48.242.22 (talk) 21:41, 8 September 2011 (UTC)

Made Revisions / Comments Welcome / Opinions Offered
I made some moderate (not major, but not typo corrections either) to the article Oct. 25-26, 2014. Comments/corrections/critiques are welcome.

I fixed some issues (IMHO, of course), but others remain. These are

1. Article title: In this situation, we're not dealing with a pure mathematics text, where every word of a definition is scrutinized, edited, and re-edited, and then re-edited again, etc. Engineering terminology and documents are inherently sloppier. I'd prefer "Differential Multilateration" to just "Multilateration". But I could live with the latter, provided the terminology confusion is explained early in the article.

2. "Trilateration". This is related to #1. Differential Multilateration can be done with many ("multi") stations, but Absolute Multilateration can only be done with three stations ("tri")? Really -- can anyone cite a source? I'll acknowledge that Differential Multilateration implementations are more common than Absolute Multilateration implementations in the aerospace application domain. But this terminology is wrong "on its face". For example, an aircraft (at sufficient altitude) can measure (a) its distance to more that two DME ground stations, plus (b) its altitude with a barometric or radio altimeter, and then (c) compute is position. The position solution can either "clamp" the altitude to the altimeter reading or be a full three-dimensional solution.

3. "GPS is not a Differential Multilateration system". This statement/position is factually incorrect. GPS *IS -- repeat, IS*, repeat ... -- a Differential Multilateration system. To not understand this calls into question every statement by the originator. — Preceding unsigned comment added by NavigationGuy (talk • contribs) 00:38, 28 October 2014 (UTC)

An extra transmitter/reciever/satellite
When going from 2 to 3 receivers there is only one extra independ TDOA available. Often it is mentioned that there are two extra TDOAs available and this is completely true but there is only one independend extra TDOA avalable. Consider the three timing signals a, b, c. The TDOA of these signals are (a-b), (b-c) and (a-c). But the (a-c) TDOA is equivalent to (a-b)+(b-c) and does not give extra information. For each extra receiver (or transmitter/satellite) there is only one extra TDOA. Using the other dependend TDOA's will not reduce the possible locations. So the curve stays the same because the extra hyperboloids intersect with the curve. Or when there are two points these dependend TDOA's do not eliminate one of the points. Crazy Software Productions (talk) 16:45, 7 November 2016 (UTC)
 * 1) One transmitter/receiver/satellite does not restrict the location at all.
 * 2) Two transmitters/receivers/satellites does gives a location on a hyperboloid.
 * 3) Three transmitters/receivers/satellites does give a location on a curve. (Intersection of two hyperboloids)
 * 4) Four transmitters/receivers/satellites gives one or two mathematical points in 3D space. (The intersection of a curve with a hyperboloid. Dependend on the constellation this gives one or two points. For GPS this is very often two points, but only one of those points is stable over time.)

Poorly Written. Second sentence was as far as I got...
The second sentence: "Unlike measurements of absolute distance or angle, measuring the difference in distance between two stations results in an infinite number of locations that satisfy the measurement." - is terrible. I don't know what this means, except that when I have to re-read something 3 times and it still makes no sense, I leave.

This entire article is poorly written. I browsed it and quickly gave up. And I'm in the GPS business. That tells you something. Someone REALLY needs to rewrite this article in proper English, so that the Average Joe doesn't give up or throw up. Either way, it's bad. 98.194.39.86 (talk) 07:39, 26 June 2017 (UTC)

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Edits Dec. 2018
1. Added a section entitled Advantages and Disadvantages

2. Added GPS, Glonass and Galileo as the most prominent examples of GNSS multilateration systems. There's some confusion on this point. Because the stations (satellites) are moving, it's less obvious that GNSSs are multilateration systems.

3. Added a section entitled Station synchronization

4. Added a section entitled User-station geometry

5. Multiple edits. A theme was to change measured TDOAs to measured TOAs and calculated TDOAs or something similar.
 * That actually what's done (how can a surveillance system with multiple remote receivers "measure" a TDOA?)
 * When the stations are fixed to the earth and the transmission are synchronized with UTC, forming TDOAs eliminates the unknown/unneeded time of transmission (TOT). However, in satellite systems are synchronized with UTC and the TOT is needed.

NavigationGuy (talk) 13:29, 16 December 2018 (UTC) NavigationGuy (talk) 13:46, 19 December 2018 (UTC) NavigationGuy (talk) 16:00, 25 December 2018 (UTC)

Suspect typo
from Correct typos in one click motiated->motivated? context: wo-dimensional Cartesian solutions ===For finding a user's location in a two dimensional (2-D) Cartesian geometry, one can adapt one of the many methods developed for 3-D geometry, most motiated motiated by GPS -- for example, Bancroft's&lt;ref name="Geyer98"&gt;"Solving Passive Multilateration Equations Using Bancroft’s Algorithm", Michael Geyer and Anastasios Daskalakis, ''Di

Shorten Intro Section
I attempted to shorten the Intro section, per the note. NavigationGuy (talk) 12:20, 8 April 2022 (UTC)